It seems natural to me to generalize the notion of perfect number to rings of integers more general that ${\Bbb Z}$. I'll want to think of number fields concretely, as subfields of ${\Bbb C}$. For a mutatis mutandis definition, a given algebraic integer must have a distinguished finite set of divisors. That makes two tricky parts: figuring out what to say if the group of units is richer than $\{-1,1\}$ making due with the order if the field is not a subfield of ${\Bbb R}$.

Do definitions occur in the literature? Do nontrivial examples exist?

It seems natural to me to generalize the notion of perfect number to rings of integers more general that ${\Bbb Z}$. I'll want to think of number fields concretely, as subfields of ${\Bbb C}$. For a mutatis mutandis definition, a given algebraic integer must have a distinguished finite set of divisors. That makes two tricky parts:

1. figuring out what to say if the group of units is richer than $\{-1,1\}$
2. making do without the order if the field is not a subfield of ${\Bbb R}$.

Do definitions occur in the literature? Do nontrivial examples exist?